## Girsanov Example

Let \(u(x)\colon \mathbf{R}^d \rightarrow \mathbf{R}^d\) such that \(\sup_x |u(x)| \leq K \). Define \(X_t\) by

\[dX_t = u(X_t)dt + \sigma dB_t\]

for \(\sigma >0\) and \(X_0=x\).

For any open set \( A \subset \mathbf{R}^d\) assume that you know that \(P(B_t \in A) >0)\) show that the same holds for \(X_t\).

Hint: Start by showing that \(\mathbf{E}[ f(x+ \sigma B_t)] = \mathbf{E}[\Lambda_t f(X_t)]\) for some process \(\Lambda_t\) and any function \(f\colon \mathbf{R}^d\rightarrow \mathbf{R}\). Next show that \(\mathbf{E}[\Lambda_t^2] < \infty\)

## Exit Through Boundary II

Consider the following one dimensional SDE.

\begin{align*}

dX_t&= \mu dt+ \sin( X_t )^\alpha dW(t)\\

X_0&=\frac{\pi}2

\end{align*}

Consider the equation for \(\alpha >0\) and \(\mu \in \mathbf{R}\). On what interval do you expect to find the solution at all times ? Classify the behavior at the boundaries in terms of the parameters.

For what values of \(\alpha < 0\) does it seem reasonable to define the process ? any ? justify your answer.

## Practice with Ito Formula

Let \(B_t\) be a standard Brownian motion. For each of the following definitions of \(Y_t\), find adapted stochastic process \(\mu_t\) and \(\sigma_t\) so that \(dY_t =\mu_t dt + \sigma_t dB_t\)

- \( Y_t =\sin(B_t) \)
- \( Y_t= (B_t)^p \) for \(p>0\)
- \( Y_t=\exp( B_t – t^2)\)
- \(Y_t=\log(B_t) \)
- \(Y_t= t^2 B_t \)

## Product Chain

Let \(Z_n\) be a collection of independent random variables with \(P(Z_n=1)=\frac12\) and \(P(Z_n=\frac12)=\frac12\) . Define \(X_0=1\) and \(X_{n+1}=Z_n X_n\).

- What is \(E( X_n | X_{n-1})\) ?
- What is \(E(X_n)\) ?
- What is \(\mathrm{Cov}(X_n,X_{n-1})\) ?

## Basic Markov Chain I

In each of the graphs pictured, assume that each arrow leaving a vertex has an equal chance of being followed. Hence if there are thee arrows leaving a vertex then there is a 1/3 chance of each being followed.

- For each of the six pictures, find the Markov transition matrix.
- State if the Markov chain given by this matrix is irreducible.
- If the Matrix is irreducible, state if it is aperiodic.

## Conditional Variance

Given two random variables, we define the conditional variance of \(X\) given \(Y\) by

\[ \text{Var}(X | Y ) = E(X^2 | Y ) – (\,E(X | Y )\,)^2 \]

Show that

\[ \text{Var}(X ) = E (\text{Var}(X | Y ) ) +\text{Var}( E(X | Y ) ) \,. \]

Of course \( E(X | Y )\) is just random variable so we have that

\[ \text{Var}( E(X | Y ) ) = E [\,E(X | Y )^2\,] – [\,E(\,E(X | Y )\,)\,]^2 \]

also is is useful to recall that \(E(E(X | Y ))= E(X )\).

## Random Walk

Let \(\{Z_1, Z_2, \dots, Z_n,\dots\} \) be a sequence of i.i.d random variables such that

\[ P( Z_k = a ) = \begin{cases} \frac14 & \text{ If } a=1 \\\frac14 & \text{ If } a=0\\\frac12 & \text{ If } a=-1 \end{cases}\]

If \(X_{n+1} = X_n + Z_n\) and \(X_0=1\) what is

- \( E( X_{n+1} | X_{n}) \)?
- \( E( X_{n+1}) \)?
- \( \text{Var}( X_{n+1} | X_{n}) \)?
- \( \text{Var}( X_{n+1} )\)?

Notice that \(X_n\) depends only \(Z_{n-1},Z_{n-2},\dots,Z_1\) and hence \(X_n\) is independent of \(Z_n\) !

## Joint Density Poisson arrival

Let \(T_1\) and \(T_5\) be the times of the first and fifth arrivals in a Poisson arrival prices with rate \(\lambda\). Find the joint distribution of \(T_1\) and \(T_5\) .

## Uniform Spacing

Let \(U_1, U_2, U_3, U_4, U_5\) be independent uniform \((0,1)\) random variables. Let \(R\) be the difference between the max and the min of the random variables. Find

- \( E( R)\)
- the joint density of the min and the max of the \(U\)’s
- \(P( R>0.5)\)

[Pitman p. 355 #14]

## Joint density part 2

Let \(X\) and \(Y\) have joint density

\(f(x,y) = 90(y-x)^8, \quad 0<x<y<1\)

- State the conditional distribution of \(X \mid Y\) and \(Y \mid X\)
- Are these two random variables independent?
- What is \( \mathbf{P}(Y \mid X=.2 ) \) and \( \mathbf{E}(Y \mid X=.2) \) ?

What is \( \mathbf{P}(Y \mid X=.2 ) \) and \( \mathbf{E}(Y \mid X=.2) \)

[Adapted from Pitman pg 354]

## Uniform distributed points given an arrival

Consider a Poisson arrival process with rate \(\lambda>0\). Let \(T\) be the time of the first arrival starting from time \(t>0\). Let \(N(s,t]\) be the number of arrivals in the time interval \((s,t]\).

Fixing an \(L>0\), define the pdf \(f(t)\) by \(f(t)dt= P(T \in dt | N(0,L]=1)\) for \(t \in (0,L]\). Show that \(f(t)\) is the pdf of a uniform random variable on the interval \([0,L]\) (independent of \(\lambda\) !).

## Minimum Dice Roll

Suppose that three fair 6-sided dice are rolled.

- Let \(M\) be the minimum of three numbers rolled. Find \(\mathbb{E}(M)\).
- Let \(S\) be the sum of the largest two rolls. Find \(\mathbb{E}(S)\).

## Dinner Party Seating

A host invites \(n\) guests to a party (guest #1, guest #2, … , guest #n). Each guest brings with them their best friend. At the party there is a large circular table with \2n\) seats. All of the \(n\) invited guests and their best friends sit in a random seat.

- What is the probability that guest #1 is seated next to their best friend?
- What is the expected number of the \(n\) invited guests who are seated next to their best friend?

## Telephone Calls throughout the Week

Telephone calls come in to a customer service hotline. The number of calls that arrive within a certain time frame follows a Poisson distribution. The average number of calls per hour depends on the day of the week. During the week (Monday through Friday) the hotline receives an average of 10 calls per hour. Over the weekend (Saturday and Sunday) the hotline receives and average of 5 calls per hour. The hotline operates for 8 hours each day of the week. (The number of calls on one day is independent of the numbers of calls on other days.)

- What is the probability that the center receives more than 500 calls in 1 week?
- Each person who calls the center has a 20% chance of getting a refund (independent of other callers). Find the probability that 10 or fewer people get a refund on Tuesday.
- One day of the week is chosen uniformly at random. On this day, a representative at the call center reports that 60 people called in. Based on that information, what is the probability that the day was a weekend day (either Saturday or Sunday)?

## A Dice Rolling Game

15 players each roll a fair 6-sided die once. If two or more players roll the same number, those players are eliminated. What is the expected number of players who get eliminated?

## January Birthdays at a Call Center

Calls arrive at a call center according to a Poisson arrival process with an average rate of 2 calls/minute. Each caller has a 1/12 chance of having a January birthday, independent of other callers. What is the expected wait time until the call center receives 3 calls from callers with January birthdays?

## Rock Paper Scissors

A game of rock paper scissors consists of several rounds (players continue to play rounds until one player wins). In one round of rock paper scissors, two players each choose one of three options (rock, paper, or scissors). If they choose two different options, the game ends (rock beats scissors, scissors beat paper, and paper beats rock). If both players choose the same option, the game continues for another round. Assume each player chooses rock, paper or scissors uniformly at random and independently.

- What is the distribution of number of rounds played in a single game?
- What is the expected number of rounds in a game? What is the standard deviation of number of rounds in a game?
- Let \(n\) be the number of games played. How big must \(n\) be to ensure at least 100 rounds are played with 90% probability? Use an appropriate approximation to estimate.

## Birds Arrive at a Bird Feeder

Birds arrive at a bird feeder according to a Poisson arrival process with a rate of 6 birds per hour. A person starts watching the feeder at time 0.

- What is the probability that the first three birds arrive at the feeder within 30 minutes?
- What is the expected time it takes for the 10th bird to arrive?
- 10% of the birds who visit the feeder are cardinals. What is the probability that the 3rd cardinal to arrive at the feeder is the 10th bird to arrive at the feeder?
- 10% of the birds who visit the feeder are cardinals. The person watching the feeder decides to continue watching the feeder until they see a cardinal. What is the probability that the person waits more than 5 hours?

## Ant crawls along a number line

An ant is crawling on a number line. The ant starts out at position \(0\). Every second the ant either

- Moves to the right 1 unit, with probability 1/2,
- Moves to the left 1 unit, with probability 1/4, or
- Stays at its current location, with probability 1/4

The ant’s movement during a particular second is independent of the ant’s previous movements. Let \(X_{160}\) be the ant’s location after 160 seconds.

- In 160 seconds, what is the probability that the ant moves to the right exactly 80 times, and to the left exactly 40 times?
- What is \(\mathbb{E}(X_{160})\)?
- What is \(Var(X_{160})\)?
- Let \(\mu=\mathbb{E}(X)\). Estimate \(\mathbb{P}(|X-\mu|\geq 15)\).

## Another Birthday Problem

A host invites guests to a party. How many guests should be invited in order for the expected number of guests who share a birthday with at least one other guest to be at least 4?

## Almost geometric

An experimenter rolls a fair 6-sided die until they’ve seen both a 1 and a 2 (not necessarily consecutively). What is the experimenter’s expected number of rolls?

## Using a Mass Function

Let \(X\) be a random variable with probability mass function

\(p(n) = \frac{1}{c^n}\quad \text{for } n=2,3,4,\cdots\)

and \(p(x)=0\) otherwise.

- Find \(c.\)
- Compute the probability that \(X\) is even.

## Ant crawls along a grid

An ant crawls along a coordinate grid. The ant starts at \((0,0)\). At each step, the ant either moves up one unit (with probability 1/2) or to the right 1 unit (with probability 1/2).

After 5 steps the ant has

- a 5/32 chance of being at the coordinate \((4,1)\),
- a 10/32 chance of being at the coordinate \((3,2)\), and
- a 17/32 chance of being at one of the coordinates \((2,3), (1,4), (5,0), (0,5)\)

What is the probability that the ant is at position \((4,2)\) after 6 steps?

## Magic Die (or is it rigged?)

A magician claims to have a magic die. If the die is rolled and lands on an even number, then the next time the die is rolled it will land on an odd number (and vice versa). So, as the die is rolled it will alternate perfectly between even and odd numbers (or so the magician claims).

You, being skeptical, figure there’s a 1 percent chance that the die is magical and a 99 percent chance that it’s just an ordinary fair die. You then ask the magician to “prove” the die is magical by rolling it some number of times.

How many successfully alternating rolls will it take for you to think there’s a 99 percent chance the die is magical (or, more likely, that it’s rigged in some way so it always alternates)?

## Identifying a Biased Coin

You have a fair coin and a biased coin, but you can’t tell which is which. The biased coin lands on heads 75% of the time. You decide to try to determine which coin is the biased coin by selecting one of the coins at random and flipping tn 100 times. Let \(\hat{p}\) be your observed fraction of heads. Based on \(\hat{p}\), you decide which coin is the biased one.

- For which values of \(\hat{p}\) will you assume the coin you flipped is the biased coin?
- What is the probability that you correctly determine which coin is the biased coin?

## Boxes with Yellow and Green Balls

Three boxes contain yellow and green balls

- Box 1 contains 2 yellow balls.
- Box 2 contains 2 green balls.
- Box 3 contains 1 yellow ball and 1 green ball.

One box is selected at random, and one ball is pulled out of that box.

- The ball that is pulled out of the chosen box is yellow. What is the probability that the other ball in that same box is also yellow?
- Let \(A\) be the event that Box 3 is chosen. Let \(B\) be the event that a yellow ball is pulled out of the chosen box. Are \(A\) and \(B\) independent?
- The ball that is pulled out of the chosen box is yellow. Without replacement, a second ball is chosen at random from one of the three boxes. (Each box has a 1/3 chance of being selected.) What is the probability that the second ball chosen is also yellow?

## Consecutive Tails

Let \(A_n\) be the event that in \(n\) flips of a fair coin, there are never 2 consecutive tails. Suppose we know the following probabilities.

- \(\mathbf{P}(A_{19})\approx 0.021\)
- \(\mathbf{P}(A_{20})\approx 0.017\)

Evaluate \(\mathbf{P}(A_{21})\)

## Joint Distribution Table

Consider the following joint distribution.

If the experiment is flipping a fair coin three times, which of the following could be the random variables \(X\) and \(Y\). Select all that apply.

- \(X=\) the number of heads, \(Y=\) the number of tails.
- \(X=\) the number of tails, \(Y=\) the number of tails (i.e., \(Y=X\)).
- \(X=\) the number of heads. \(Y=3-X.\)
- \(X=\) the number of tails on the first two flips. \(Y=\) the number of tails on the last two flips.

## Hair and Eye Color

About 60% of the world’s population has brown eyes. About 20% of the world’s population has brown hair. Given that a person has brown eyes, they have a 10% chance of also having brown hair.

Given that a randomly selected person does not have brown eyes what is the probability that they also do not have brown hair?

## A strange deck of cards

In a non-standard deck of cards there are

- 20 blue cards (numbered 1 through 20),
- 20 green cards (numbered 1 through 20), and

20 red cards (numbered 1 through 20)

Four cards are dealt without replacement from this deck.

- What is the probability that exactly two of the four cards dealt are blue?
- Given that at least one of the first two cards dealt is blue, what is the probability that exactly three of the four cards dealt are blue?
- What is the probability that at least two of the four cards dealt have the same numeric value (1 through 20)?